lmomTLgld {lmomco} | R Documentation

## Trimmed L-moments (t=1) of the Generalized Lambda Distribution

### Description

This function estimates the symmetrical trimmed L-moments (TL-moments) for t=1 of the Generalized Lambda distribution given the parameters (xi, α, kappa, and h) from `vec2par`. The TL-moments in terms of the parameters are complicated; however, there are analytical solutions. There are no simple expressions of the parameters in terms of the L-moments. The first four TL-moments of the distribution are

λ^{(1)}_1 = xi + 6α (frac{1}{(kappa+3)(kappa+2)} - frac{1}{(h+3)(h+2)} ) mbox{,}

λ^{(1)}_2 = 6α (frac{kappa}{(kappa+4)(kappa+3)(kappa+2)} + frac{h}{(h+4)(h+3)(h+2)}) mbox{,}

λ^{(1)}_3 = frac{20α}{3} (frac{kappa (kappa - 1)} {(kappa+5)(kappa+4)(kappa+3)(kappa+2)} - frac{h (h - 1)} {(h+5)(h+4)(h+3)(h+2)} ) mbox{,}

λ^{(1)}_4 = frac{15α}{2} (frac{kappa (kappa - 2)(kappa - 1)} {(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} + frac{h (h - 2)(h - 1)} {(h+6)(h+5)(h+4)(h+3)(h+2)} ) mbox{, and}

λ^{(1)}_5 = frac{42α}{5} (N1 - N2 ) mbox{,}

where

N1 = frac{kappa (kappa - 3)(kappa - 2)(kappa - 1) } {(kappa+7)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} mbox{ and}

N2 = frac{h (h - 3)(h - 2)(h - 1)}{(h+7)(h+6)(h+5)(h+4)(h+3)(h+2)} mbox{.}

The TL-moment (t=1) for tau^{(1)}_3 is

tau^{(1)}_3 = frac{10}{9} ( frac{kappa(kappa-1)(h+5)(h+4)(h+3)(h+2) - h(h-1)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} {(kappa+5)(h+5) times [kappa(h+4)(h+3)(h+2) + h(kappa+4)(kappa+3)(kappa+2)] } ) mbox{.}

The TL-moment (t=1) for tau^{(1)}_4 is

N1 = kappa(kappa-2)(kappa-1)(h+6)(h+5)(h+4)(h+3)(h+2) mbox{,}

N2 = h(h-2)(h-1)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2) mbox{,}

D1 = (kappa+6)(h+6)(kappa+5)(h+5) mbox{,}

D2 = [kappa(h+4)(h+3)(h+2) + h(kappa+4)(kappa+3)(kappa+2)] mbox{, and}

tau^{(1)}_4 = frac{5}{4} ( frac{N1 + N2}{D1 times D2} ) mbox{.}

Finally the TL-moment (t=1) for tau^{(1)}_5 is

N1 = kappa(kappa-3)(kappa-2)(kappa-1)(h+7)(h+6)(h+5)(h+4)(h+3)(h+2) mbox{,}

N2 = h(h-3)(h-2)(h-1)(kappa+7)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2) mbox{,}

D1 = (kappa+7)(h+7)(kappa+6)(h+6)(kappa+5)(h+5) mbox{,}

D2 = [kappa(h+4)(h+3)(h+2) + h(kappa+4)(kappa+3)(kappa+2)] mbox{, and}

tau^{(1)}_5 = frac{7}{5} ( frac{N1 - N2}{D1 times D2} )mbox{.}

By inspection the tau_r equations are not applicable for negative integer values k={-2, -3, -4, ... } and h={-2, -3, -4, ... } as division by zero will result. There are additional, but difficult to formulate, restrictions on the parameters both to define a valid Generalized Lambda distribution as well as valid L-moments. Verification of the parameters is conducted through `are.pargld.valid`, and verification of the L-moment validity is conducted through `are.lmom.valid`.

### Usage

    lmomTLgld(gldpara)

### Arguments

`gldpara` | The parameters of the distribution.

### Value

An R `list` is returned.

`lambdas` | Vector of the TL-moments. First element is λ^{(1)}_1, second element is λ^{(1)}_2, and so on.
`ratios` | Vector of the TL-moment ratios. Second element is tau^{(1)}, third element is tau^{(1)}_3 and so on.
`trim` | Trim level = 1
`source` | An attribute identifying the computational source of the TL-moments: “lmomTLgld”.

### Author(s)

W.H. Asquith

### Source

Derivations conducted by W.H. Asquith on February 18 and 19, 2006.

### References

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational statistics and data analysis, vol. 43, pp. 299-314.

Hosking, J.R.M., 1990, L-moments–Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105-124.

Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distributions–The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.

### See Also

`lmomgld`, `pargld`, `cdfgld`, `quagld`

### Examples

    lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))

[Package lmomco version 0.92 Index]